Cartesian Product
The Cartesian product of two sets A and B, denoted as A × B, is a set that contains all possible ordered pairs where the first element of each pair is from set A and the second element is from set B.
Formally, if A = {a1, a2, a3, ...} and \; B = {b1, b2, b3, ...}, then the Cartesian product A × B is defined as:
A \times B={(a_1, b_1), (a_1, b_2), \ldots, (a_2, b_1), (a_2, b_2), \ldots, (a_3, b_1), (a_3, b_2), \ldots, }
So, it’s a set of all possible combinations of elements from A and B, where each element from A is paired with every element from B.
Relation
A relation is a set of ordered pairs of elements from one set to another.
A relation R between two sets A and B can be formally defined as follows:
R \subseteq A \times B
Types of Relation
1. Reflexive relation
A reflexive relation is a specific type of relation, where every element in the set is related to itself. In other words, a relation R on a set A is reflexive if for every element ‘a’ in set A, the ordered pair (a, a) belongs to the relation R.
Formally, a relation R on a set A is reflexive if, for all elements a in A:
A relation R on a set A is reflexive if, for all elements a in A:
\forall a \; \in A \; [ (a, a) \in R ]
In simpler terms, if you have a set A and a relation R, and for every element ‘a’ in set A, the pair (a, a) is in R, then R is considered a reflexive relation.
For example, consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3)}. In this case, R is a reflexive relation because it includes all the pairs (a, a) for each element a in A, demonstrating the property of reflexivity.
2. Symmetric Relations
In mathematics, a symmetric relation is a specific type of relation on a set where if an element ‘a’ is related to an element ‘b’, then ‘b’ is also related to ‘a’. In other words, if (a, b) is in the relation, then (b, a) must also be in the relation.
Formally, a relation R on a set A is symmetric if, for all elements ‘a’ and ‘b’ in A:
[ (a, b) \in R \Rightarrow (b, a) \in R ]
In simpler terms, in a symmetric relation, if there is a connection or relationship between two elements ‘a’ and ‘b’, it implies that there is also a relationship between ‘b’ and ‘a’ in the same relation.
For example, consider a set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1), (2, 3), (3, 2)}. In this case, R is a symmetric relation because for every pair (a, b) in R, the corresponding pair (b, a) is also present in R, satisfying the symmetry property.
3. Transitive Relation
A transitive relation is a specific type of relation on a set where if an element ‘a’ is related to another element ‘b’, and ‘b’ is related to a third element ‘c’, then it must also be the case that ‘a’ is related to ‘c’. In other words, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.
Formally, a relation R on a set A is transitive if, for all elements ‘a,’ ‘b,’ and ‘c’ in A:
[ (a, b) \in R \text{ and } (b, c) \in R \Rightarrow (a, c) \in R ]
In simpler terms, in a transitive relation, if there’s a chain of relationships between elements, it implies that there’s a direct relationship between the endpoints of that chain.
For example, consider a set A = {1, 2, 3} and a relation R = {(1, 2), (2, 3), (1, 3)}. In this case, R is a transitive relation because for every pair (a, b) and (b, c) in R, the corresponding pair (a, c) is also present in R, satisfying the transitive property.
4. Equivalence Relation
A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
Equivalence Class
An equivalence class is a subset of a set that contains all elements that are considered equivalent or indistinguishable under an equivalence relation.
Equivalence classes are used to partition a set into groups or subsets of elements that share a common equivalence with each other.
Formally, let R be an equivalence relation on a set A. For any element ‘a’ in A, the equivalence class [a] (read as “the equivalence class of ‘a'”) is defined as follows:
[ [a] = \{ x \in A \,|\, (a, x) \in R \} ]
In simpler terms, the equivalence class [a] consists of all elements ‘x’ in the set A for which the pair (a, x) is related by the equivalence relation R. In other words, all elements in the equivalence class [a] are equivalent to ‘a’ according to the relation R.
5. Empty Relation
An empty relation, often referred to as the empty set or the void relation, is a specific type of binary relation on a set where there are no ordered pairs present in the relation. In other words, it is a relation that has no elements.
Formally, the empty relation on a set A, denoted as ∅ or {}, is defined as follows:
\emptyset = \{\} = \{\text{No ordered pairs in the relation}\}In this relation, there are no pairs of elements from set A that belong to the relation. It represents a situation where there are no relationships or connections between any elements in the set A.
For example, if we have a set A = {1, 2, 3}, the empty relation on A would be represented as ∅ or {}. It means that there are no ordered pairs (a, b) in the relation where ‘a’ and ‘b’ are elements of set A.
The empty relation is the opposite of a universal relation.
6. Universal Relation
In mathematics, a universal relation, also known as a universal set relation, is a binary relation on a set that includes all possible ordered pairs of elements from that set. In other words, it relates every element in the set to every other element in the set.
Formally, if A is a set, the universal relation U on set A is defined as:
U = A \times A \
In essence, the universal relation U includes every possible pair of elements (a, b) where ‘a’ and ‘b’ both belong to set A. It’s a way of expressing that every element in the set is related to every other element in the set.
The universal relation is often used as a theoretical construct and can be helpful in various mathematical and set-theoretical contexts.